Group #34005432 — subgroups

Subgroups — #34005432

Order 18688.

342 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625864	C₂	1	1	yes	maximal, characteristic
2	#22625846	C₂	1	1	yes	maximal, characteristic
3	#22624761	C₂	1	4	yes	maximal
4	#22609675	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556658	—	2	2	no	—
6	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
7	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
8	#17989453	—	2	2	no	—
9	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
10	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
11	#17988342	C₄	1	1	yes	characteristic
12	#17988341	C₄	1	2	yes	—
13	#336×C₇₃	C₄	1	1	yes	characteristic
14	C₂²×#10596833	—	2	2	no	—
15	C₂²×#10596833	—	2	2	no	—
16	#14556658	—	4	4	no	—
17	C₄⋊#10596833	—	2	4	no	—
18	C₄⋊#10596833	—	2	4	no	—
19	C₂²×#10596831	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
22	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
23	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
24	C₈×Dic₇₃	—	2	2	no	—
25	C₈×Dic₇₃	—	2	2	no	—
26	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#10596833	—	4	2	no	—
29	C₂×#10596833	—	4	2	no	—
30	C₂×#10596833	—	4	2	no	—
31	C₂×#10596833	—	4	4	no	—
32	C₂×#10596833	—	4	2	no	—
33	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
34	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×#10596831	—	2	2	no	—
37	C₂×#10596831	—	2	2	no	—
38	C₂×#10596831	—	2	2	no	—
39	C₄×Dic₇₃	—	2	1	no	—
40	C₄×Dic₇₃	—	2	1	no	—
41	C₄×Dic₇₃	—	2	4	no	—
42	C₄×Dic₇₃	—	2	2	no	—
43	C₄×Dic₇₃	—	2	2	no	—
44	C₄×Dic₇₃	#41	1	2	yes	—
45	C₄×Dic₇₃	#41	1	2	yes	—
46	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₃	—	2	2	no	—
49	C₂×C₈×C₇₃	—	2	2	no	—
50	C₂×C₈×C₇₃	—	2	2	no	—
51	C₄²×C₇₃	—	2	2	no	—
52	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	—	2	2	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	#117	1	2	yes	—
57	C₂×C₄×C₇₃	#131	1	2	yes	—
58	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
60	C₂×Dic₇₃	—	2	2	no	—
61	C₂×Dic₇₃	—	2	2	no	—
62	C₂×Dic₇₃	—	2	2	no	—
63	C₂×Dic₇₃	#108	1	4	yes	—
64	C₂×Dic₇₃	#124	1	1	yes	characteristic
65	C₂×Dic₇₃	#124	1	1	yes	characteristic
66	C₂×Dic₇₃	#125	1	2	yes	—
67	C₂×Dic₇₃	AGL(1,ℤ/8)	1	2	yes	—
68	C₂×Dic₇₃	#137	1	2	yes	—
69	#10596833	—	8	2	no	—
70	#10596833	—	8	2	no	—
71	C₈×C₇₃	—	4	2	no	—
72	#10596831	—	4	2	no	—
73	C₂²×C₇₃	#382	1	1	yes	characteristic
74	C₂²×C₇₃	#424	1	1	yes	characteristic
75	C₂²×C₇₃	#424	1	1	yes	characteristic
76	C₂²×C₇₃	#428	1	2	yes	—
77	C₂²×C₇₃	C₂×#108	1	2	yes	—
78	C₄×C₇₃	—	4	2	no	—
79	C₄×C₇₃	#433	1	2	yes	—
80	C₄×C₇₃	#433	1	2	yes	—
81	Dic₇₃	—	2	4	no	—
82	Dic₇₃	—	2	2	no	—
83	Dic₇₃	#489	1	2	yes	—
84	Dic₇₃	#490	1	1	yes	characteristic
85	Dic₇₃	#491	1	1	yes	characteristic
86	#8789	—	73	1	no	maximal, Sylow
87	C₂×C₇₃	#1573	1	2	yes	—
88	C₂×C₇₃	#1574	1	2	yes	—
89	C₂×C₇₃	#1832	1	1	yes	characteristic
90	C₂×C₇₃	#1836	1	1	yes	characteristic
91	C₂×C₇₃	#1837	1	1	yes	characteristic
92	#1842	—	73	1	no	—
93	#1831	—	73	1	no	—
94	#1744	—	73	1	no	—
95	#1277	—	73	4	no	—
96	C₇₃	#8789	1	1	yes	characteristic, complemented, Sylow
97	C₂×C₄⋊C₈	—	73	2	no	—
98	C₂×C₄⋊C₈	—	73	2	no	—
99	C₂×C₄⋊C₈	—	146	2	no	—
100	C₂×C₄×C₈	—	73	2	no	—
101	C₄³	—	73	1	no	—
102	#336	—	73	2	no	—
103	#336	—	73	1	no	—
104	#336	—	146	2	no	—
105	#336	#64408	1	1	yes	characteristic
106	C₂²×C₈	—	73	2	no	—
107	C₂²×C₈	—	146	2	no	—
108	C₂²×C₈	—	146	2	no	—
109	C₂²×C₈	C₂×#64408	1	2	yes	—
110	C₂×C₄²	—	73	1	no	—
111	C₂×C₄²	—	73	2	no	—
112	C₂×C₄²	—	73	2	no	—
113	C₂×C₄²	—	73	1	no	—
114	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
115	C₄⋊C₈	—	292	4	no	—
116	C₄⋊C₈	—	146	4	no	—
117	C₄⋊C₈	—	146	4	no	—
118	C₄×C₈	—	146	2	no	—
119	C₄×C₈	—	146	2	no	—
120	C₂²×C₄	—	73	1	no	—
121	C₂²×C₄	—	73	2	no	—
122	C₂²×C₄	—	73	1	no	—
123	C₂²×C₄	C₄⋊#64408	1	2	yes	—
124	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	292	4	no	—
129	C₂×C₈	—	292	2	no	—
130	C₂×C₈	—	292	2	no	—
131	C₂×C₈	—	292	2	no	—
132	C₂×C₈	—	292	2	no	—
133	C₂×C₈	—	146	2	no	—
134	C₂×C₈	—	146	2	no	—
135	C₂×C₈	—	146	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	73	2	no	—
138	C₄²	—	73	2	no	—
139	C₄²	—	146	2	no	—
140	C₄²	—	146	2	no	—
141	C₄²	—	146	4	no	—
142	C₄²	—	146	1	no	—
143	C₄²	—	146	1	no	—
144	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	73	1	no	—
149	C₂×C₄	—	73	1	no	—
150	C₂×C₄	—	73	2	no	—
151	C₂×C₄	—	73	2	no	—
152	C₂×C₄	—	73	2	no	—
153	C₂×C₄	—	73	4	no	—
154	C₂×C₄	—	146	2	no	—
155	C₂×C₄	—	146	2	no	—
156	C₂×C₄	—	146	2	no	—
157	C₂×C₄	#14556670	1	2	yes	—
158	C₂×C₄	#14556678	1	1	yes	characteristic
159	C₂×C₄	#14556678	1	2	yes	—
160	C₂×C₄	#14556679	1	1	yes	characteristic
161	C₈	—	4	2	no	—
162	C₈	—	292	2	no	—
163	C₈	—	584	2	no	—
164	C₈	—	584	2	no	—
165	C₂²	#17989531	1	1	yes	characteristic
166	C₂²	#17989593	1	1	yes	characteristic
167	C₂²	#17989594	1	1	yes	characteristic
168	C₂²	#17989598	1	2	yes	—
169	C₂²	D₇₃⋊#108	1	2	yes	—
170	C₄	—	4	2	no	—
171	C₄	—	73	1	no	—
172	C₄	—	73	2	no	—
173	C₄	—	73	1	no	—
174	C₄	—	146	2	no	—
175	C₄	—	146	4	no	—
176	C₄	#17989611	1	2	yes	—
177	C₄	#17989612	1	2	yes	—
178	C₂	#22625505	1	2	yes	—
179	C₂	#22625507	1	2	yes	—
180	C₂	#22625849	1	1	yes	characteristic
181	C₂	#22625854	1	1	yes	characteristic
182	C₂	#22625856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625864

C₂

yes

maximal, characteristic

#22625846

C₂

yes

maximal, characteristic

#22624761

C₂

yes

maximal

#22609675

C₂

yes

maximal, characteristic

C₂×#14556658

—

C₂×C₄⋊#10596833

C₂²

yes

—

C₂×C₄⋊#10596833

C₂²

yes

—

#17989453

—

C₂×C₈×Dic₇₃

C₂²

yes

—

C₄²×Dic₇₃

C₂²

yes

characteristic

#17988342

C₄

yes

characteristic

#17988341

C₄

yes

—

#336×C₇₃

C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₇₃

D₄

yes

—

C₂×C₄×Dic₇₃

Q₈

yes

—

C₂×C₄×Dic₇₃

C₂³

yes

characteristic

C₈×Dic₇₃

—

C₈×Dic₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

—

—

—

—

C₂²×Dic₇₃

C₄⋊C₄

yes

—

C₂²×Dic₇₃

C₂×D₄

yes

characteristic

C₂²×Dic₇₃

C₂×Q₈

yes

—

—

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

#41

yes

—

C₄×Dic₇₃

#41

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

#131

yes

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#108

yes

—

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

C₂×Dic₇₃

yes

—

C₂×Dic₇₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₃

#137

yes

—

#10596833

—

#10596833

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

C₂×#108

yes

—

C₄×C₇₃

—

C₄×C₇₃

#433

yes

—

C₄×C₇₃

#433

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#489

yes

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

—

C₂×C₇₃

#1574

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

—

C₇₃

#8789

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

146

—

100

C₂×C₄×C₈

—

101

C₄³

—

102

#336

—

103

#336

—

104

#336

—

146

—

105

#336

#64408

yes

characteristic

106

C₂²×C₈

—

107

C₂²×C₈

—

146

—

108

C₂²×C₈

—

146

—

109

C₂²×C₈

C₂×#64408

yes

—

110

C₂×C₄²

—

111

C₂×C₄²

—

112

C₂×C₄²

—

113

C₂×C₄²

—

114

C₂×C₄²

C₂×#64408

yes

characteristic

115

C₄⋊C₈

—

292

—

116

C₄⋊C₈

—

146

—

117

C₄⋊C₈

—

146

—

118

C₄×C₈

—

146

—

119

C₄×C₈

—

146

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₄⋊#64408

yes

—

124

C₂²×C₄

C₂²×#64408

yes

characteristic

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

292

—

129

C₂×C₈

—

292

—

130

C₂×C₈

—

292

—

131

C₂×C₈

—

292

—

132

C₂×C₈

—

292

—

133

C₂×C₈

—

146

—

134

C₂×C₈

—

146

—

135

C₂×C₈

—

146

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

146

—

140

C₄²

—

146

—

141

C₄²

—

146

—

142

C₄²

—

146

—

143

C₄²

—

146

—

144

C₂³

C₂×C₄⋊#64408

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

146

—

155

C₂×C₄

—

146

—

156

C₂×C₄

—

146

—

157

C₂×C₄

#14556670

yes

—

158

C₂×C₄

#14556678

yes

characteristic

159

C₂×C₄

#14556678

yes

—

160

C₂×C₄

#14556679

yes

characteristic

161

C₈

—

162

C₈

—

292

—

163

C₈

—

584

—

164

C₈

—

584

—

165

C₂²

#17989531

yes

characteristic

166

C₂²

#17989593

yes

characteristic

167

C₂²

#17989594

yes

characteristic

168

C₂²

#17989598

yes

—

169

C₂²

D₇₃⋊#108

yes

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

146

—

175

C₄

—

146

—

176

C₄

#17989611

yes

—

177

C₄

#17989612

yes

—

178

C₂

#22625505

yes

—

179

C₂

#22625507

yes

—

180

C₂

#22625849

yes

characteristic

181

C₂

#22625854

yes

characteristic

182

C₂

#22625856

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#64408
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#336×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	#336	#64408
hypocenter	C₇₃	#8789
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×C₄⋊#64408

commutator

C₂²×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#64408

Fitting

#336×C₇₃

C₄

socle, prime radical

C₂³×C₇₃

C₂×C₄⋊C₄

cosocle

C₂×C₄×Dic₇₃

C₂³

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
96. C₇₃	#8789	semidirect

H ≅ G/N

type

96. C₇₃

#8789

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22625864	2
#22625846	2
#22624761	2
#22609675	2
#8789	73

subgroup

index

Subgroups — #34005432

Conjugacy classes of subgroups

Complements

Maximal subgroups